Advances in mathematical economics

Raynaud de Fitte Abstract We study the integration of fuzzy level sets associated with a fuzzy random variable when the underlying space is a separable Banach space or aweak star dual of

Norimichi Hirano

Yokohama NationalUniversity

Seiichi Iwamoto

Kyushu UniversityFukuoka, JAPAN

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Kyoto UniversityKyoto, JAPAN

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Printed on acid-free paper

On the Integration of Fuzzy Level Sets 1

Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang,

and P Raynaud de Fitte

A Theory for Estimating Consumer’s Preference from Demand 33

Yuhki Hosoya

Least Square Regression Methods for Bermudan Derivatives

and Systems of Functions 57

Shigeo Kusuoka and Yusuke Morimoto

Discrete Time Optimal Control Problems on Large Intervals 91

Alexander J Zaslavski

Index 137

v

Adv Math Econ 19, 1–32 (2015)

On the Integration of Fuzzy Level Sets

Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang,

and P Raynaud de Fitte

Abstract We study the integration of fuzzy level sets associated with a fuzzy

random variable when the underlying space is a separable Banach space or aweak star dual of a separable Banach space In particular, the expectation and theconditional expectation of fuzzy level sets in this setting are presented We provethe SLLN for pairwise independent identically distributed fuzzy convex compactvalued level sets through the SLLN for pairwise independent identically distributedconvex compact valued random set in separable Banach space Some convergenceresults for a class of integrand martingale are also presented

2 C Castaing et al.

Keywords Conditional expectation • Fuzzy convex • Fuzzy martingale •

Integrand martingale • Level set • Upper semicontinuous

Article type: Research Article

In this paper we present a study of a class of random fuzzy variables whoseunderlying space is a separable Banach space E or a weak star dual Es of aseparable Banach space

The paper is organized as follows In Sect.2we summarize and state the neededmeasurable results in the weak star dual of a separable Banach space In particular,

we present the expectation and the conditional expectation of convex weak starcompact valued Gelfand-integrable mappings In Sect.3 we present the properties

of random fuzzy convex upper semi continuous integrands (variables) in Es InSect.4, the fuzzy expectation and the fuzzy conditional expectation for randomfuzzy convex upper semi continuous variables are provided in this setting Section5

is devoted to the SLLN for fuzzy convex compact (compact) valued random levelsets through the SLLN for convex compact (compact) valued random sets Theabove results lead to a new class of integrand martingales that we develop in Sect.6.Some convergence results for integrand martingales are provided

Our paper provides several issues in Fuzzy set theory, but captures different toolsfrom Probability and Set-Valued Analysis and shows the relations among them with

a comprehensive concept

in a Dual Space

Throughout this paper, ;F; P / is a complete probability space, E is a Banach

space which we generally assume to be separable, unless otherwise stated, D1 D.ek/k2N is a dense sequence in the closed unit ball of E, E is the topological

On the Integration of Fuzzy Level Sets 3

dual of E, and BE (resp BE) is the closed unit ball of E (resp E) We denote

by cc.E/ (resp cwk.E/) (resp ck.E/) the set of nonempty closed convex (resp.

weakly compact convex) (resp compact convex) subsets of E Given C 2 cc.E/,the support function associated with C is defined by

ı.x; C /D supf< x; y >; y2 C g x2 E/:

We denote by dH the Hausdorff distance on cwk.E/ A cc.E/-valued mapping

C W  ! cc.E/ is F-measurable if its graph belongs to F ˝ B.E/, where B.E/

is the Borel tribe of E For any C 2 cc.E/, we set

cwk.E/.F/ is bounded if the sequence jXnj/n2 Nis bounded in L1R.F/.

AF-measurable closed convex valued multifunction X W  ! cc.E/ is integrable

if it admits an integrable selection, equivalently if d.0; X / is integrable

We denote by Es, (resp Eb), (resp Ec) the vector space Eendowed withthe topology  E; E/ of pointwise convergence, alias w-topology (resp thetopology sassociated with the dual norm jj:jjE

b), (resp the topology cof compactconvergence) and by Em the vector space Eendowed with the topology m D

 E; H /, where H is the linear space of E generated by D1, that is the Hausdorfflocally convex topology defined by the sequence of semi-norms

We assume from now on that dE

m is held fixed Further, we have m  w 

c  s: On the other hand, the restrictions of m, w, c to any bounded subset

of Ecoincide and the Borel tribesB.E

s/ be the set of all nonempty convexweak compact subsets in E A K-valued multifunction (alias mapping forshort) X W   E

s is scalarlyF-measurable if, 8x 2 E, the support function

ı.x; X.:// is F-measurable, hence its graph belongs to F ˝ B.E

s/ Indeed,let fk/k2N be a sequence in E which separates the points of E, then we have

is F-measurable, that is, XG 2 F, this is a consequence of the Projection

Theorem (see e.g [8, Theorem III.23]) and of the equality

Consequently, the graph of u belongs to F ˝ B.E

s/ LetB be a sub--algebra

ofF It is easy and classical to see that a mapping u W  ! E

s is B; B.E

s//measurable iff it is scalarlyB-measurable A mapping u W  ! E

s is said to bescalarly integrable (alias Gelfand integrable), if, for every x 2 E, the scalar function

! 7! hx; u.!/i is F-measurable and integrable We denote by G1

E ŒE.F/ thespace of all Gelfand integrable mappings and by L1

E ŒE.F/ the subspace of all

Gelfand integrable mappings u such that the function juj W ! 7! jju.!/jjE

cwk.Es/.F/ for short) we denote the subspace of

all cwk.Es/-valued scalarly integrable and integrably bounded mappings X , that

is, such that the function jX j W ! ! jX.!/j is integrable, here jX.!/j WDsupy 2X.!/jjyjjE

b, by the above consideration, it is easy to see that jX j is

On the Integration of Fuzzy Level Sets 5

LetB be a sub--algebra of F and let X be a K-valued integrably boundedrandom set, let us define

X.!/2 Kfor all ! 2  and if X is scalarlyF-measurable We will show that

K-valued random sets enjoy good measurability properties

Proposition 1 Let X W  ! cwk.E

s/ be a convex weak-compact valued mapping The following are equivalent:

(a) XV 2 F for all m-open subset V of E.

Proof. a/ ) b/ Recall that any K 2 Kis m-compact and m  w andthe Borel tribesB.E

s/ and B.E

m / are equal Recall also that Em is a Lusinmetrizable space By (a), X is an m-compact valued measurable mapping from into the Lusin metrizable space Em Hence Graph.X / 2F ˝ B.E

m / becauseGraph.X / D f.!; x/2   E

m  W dE m.x; X.!//D 0gand the mapping !; x/7! dE

m.x; X.!// isF ˝ B.E

m /-measurable b/ ) a/ is obtained by applying the measurable Projection Theorem (seee.g [8, Theorem III.23]) and the equality

6 C Castaing et al.

Corollary 1 LetX W  ! cwk.E

s/ be a convex weak-compact valued mapping The following are equivalent:

(a) XV 2 F for all w-open subset V of E.

(b) Graph.X / 2 F ˝ B.E

s/.

(c) X admits a countable dense set of F; B.E

s//-measurable selections (d) X is scalarly F-measurable.

Proof. a/ ) d / is easy The implications d / ) b/, b/ ) c/, c/ ).d /, b/ ) a/ are already known For further details on these facts, consult

Let Xn/n2Nbe a sequence of w-closed convex sets, the sequential weakupper

limit w-ls Xnof Xn/n2Nis defined by

w-ls XnD fx2 EW xD .E; E/- lim

j !1xjI x

j 2 Xnjg:

Similarly the sequential weaklower limit w-li Xnof Xn/n2 Nis defined by

w-li XnD fx2 EW xD .E; E/- lim

n!1xnI x

n 2 Xng:

The sequence Xn/n2 Nweak star (wK for short) converges to a w-closed convex

set X1if the following holds

We need the following definition

Definition 1 The Banach space E is weakly compactly generated (WCG) if there

exists a weakly compact subset of E whose linear span is dense in E

Every separable Banach space is WCG, and every dual of a separable Banach space(endowed with the dual norm) is WCG

For the sake of completeness we recall the following [11]

Theorem 1 Suppose E is WCG (not necessarily separable) and let C and Cn.nD

1; 2; : : :/ be weak-closed, bounded, convex non empty sets ofE.

Thenı.:; Cn/ ! ı.:; C / pointwise on E if and only if the sequence Cn/ is uniformly bounded with wK limit C

Now we provide some applications

On the Integration of Fuzzy Level Sets 7

Theorem 2 LetX; Xn.n 2 N/ be a sequence in L1

cwk.E

s /.F/ with the following property:jXj C jXnj  g for all n 2 N where g is positive integrable Then the following hold:

(a) R

XdP;R

XndP , n 2 N/, are convex weak-compact,

(b) IfXnwK converges to X , equivalently, ı.:; Xn/! ı.:; X / pointwise on E, then

s ; L1E/-compact [7, Corollary 6.5.10], so thatR

XdP ,R

XndP , n D 1; 2; : : :/, are convex weak-compact

(b) Further by the Strassen theorem [8], we have that

multivalued integral of a cwk.Es/-valued mapping X˛2 L1

Then the mapping˛7!R

X.:; ˛/dP from 0; 1 into cwk.Es/ is scalarly left continuous.

8 C Castaing et al.

Proof Follows the lines of the proof of Theorem2 By (i) jX˛j  g for all ˛ 20; 1with X˛ WD X.:; ˛/ Let ˛n ! ˛ Then X.!; ˛n/ scalarly converges to X.!; ˛/,that is, ı.x; X.!; ˛n// ! ı.x; X.!; ˛// for every x 2 E and for every ! 2 .Remember thatR

X˛dP is convex weakly compact for every ˛ 20; 1 Further byStrassen’s theorem, we have

Thanks to measurable properties developed in Sect.2we present now some cations to a special class of random upper semicontinuous integrands (variables)

appli-We recall some definitions that are borrowed from the study of normal integrands(alias random lower semi-continuous integrands) on a general locally convex

Suslin space E A random lower semicontinuous (resp upper semicontinuous) integrand is a F ˝ B.E/-measurable function X defined on   E such that

X.!; :/ is lower semicontinuous (resp upper semicontinuous) The study of randomlower semicontinuous integrands occurs in some problems in Convex Analysis andVariational convergence See e.g [28] and the references therein In the following

we will focus on a special class of random upper semicontinuous integrands Herethe terminologies are borrowed from the theory of fuzzy sets initiated by Zadeh [30]and random fuzzy sets initiated by Feron [10] and Puri-Ralescu [25] According to[30] a fuzzy convex upper semicontinuous variable is a mapping X W E ! Œ0; 1

such that

(i) X is upper semicontinuous,

(ii) fx 2 E W X.x/ D 1g 6D ;

(iii) X is fuzzy convex, that is, X.x C 1  /y/  min.X.x/; X.y//, for all

2 Œ0; 1 and for all x; y 2 E

A random fuzzy convex upper semicontinuous variable is an F ˝B.E/-measurable

mapping X W   E ! Œ0; 1 such that each ! 2 , a mapping X! W E ! Œ0; 1 is

On the Integration of Fuzzy Level Sets 9

a fuzzy convex upper semicontinuous variable By upper semicontinuity and fuzzyconvexity, for each ! 2  and for each ˛ 20; 1, the level set

X˛.!/WD L˛.X /.!/WD fx 2 E W X!.x/ ˛g

is closed convex It is clear that the graph of this multifunction belongs to

F ˝ B.E/ In particular, this mapping is F-measurable by the measurable

projection theorem [8, Theorem III 23] Further, thanks to [8, Lemma III.39],for anyF ˝ B.E/-measurable mapping ' W   E ! Œ0; C1, the function

m.!/WD supf'.!; x/ W x 2 L˛.X /.!/g is F-measurable Similarly the graph ofthe multifunction fx 2 E W X.!; x/ > 0g WD ŒX! > 0 belongs toF ˝ B.E/.

In particular, this mapping isF-measurable by the measurable projection theorem

[8, Theorem III 23] Further, thanks to [8, Lemma III.39], for any F ˝

B.E/-measurable mapping ' W   E ! Œ0; C1, the function ! 7! supf'.!; x/ W

x 2 ŒX! > 0g is F-measurable In particular, if the underlying space E is aseparable Banach space, then ŒX! > 0 D fx 2 E W X.!; x/ > 0g is F-

measurable and so is the mapping ŒX!> 0 so that the mapping ! 7! supfjjxjj W

x 2 ŒX!> 0g is F-measurable, further assume that ŒX! > 0 is compact and

!7! g.!/ WD supfjjxjj W x 2 ŒX! > 0g integrable, then L˛.X / is convex compactvalued and integrably bounded: jL˛.X /j  g for all ˛ 20; 1, here measurability

of g is ensured because of the above measurable properties Similarly, for each

˛ 2 Œ0; 1Œ, L˛C.X /.!/ WD ŒX!> ˛ is compact valued and F-measurable The

above considerations still hold when the underlying space is the weak star dual Es

of a separable Banach space E because Esis a Lusin space, by measurability resultsdeveloped in Sect.2 Now we present some convergence properties of the level setsassociated with a random fuzzy convex upper semicontinuous integrand

Proposition 2 LetX W E ! Œ0; 1 random fuzzy convex upper semicontinuous integrand with the following properties:

(1) fx 2 E W X.!; x/ > 0g is compact, for each ! 2 ,

L˛.X /dP is scalarly left continuous on

10 C Castaing et al.

is convex compact,1where S1

L ˛ X /denotes the set of all integrable selections of theconvex compact valued multifunction L˛.X / We only sketch the proof See [2,8]for details Indeed, S1

L ˛ X /is convex weakly compact in L1

Eso thatR

L˛.X /dP isconvex weakly compact in E Making use of Strassen’s formula we have

On the Integration of Fuzzy Level Sets 11

Now we proceed to the study of the expectation and conditional expectation of thelevel sets associated with random fuzzy convex upper semicontinuous integrands.The following lemma is crucial for this purpose Compare with related results byPuri-Ralescu [25] dealing with fuzzy sets onRd

Lemma 1 Let X be a random fuzzy convex upper semicontinuous integrand X W

12 C Castaing et al.for all x 2 E, so thatR

X˛dP D C by the separability of E, that is,

Theorem 4 Let X be a random fuzzy convex upper semicontinuous integrand XW

X˛.!/ is scalarly left continuous on 0; 1.

Then the following holdR

a random fuzzy convex upper semicontinuous integrand defined on the dual space

Es For this purpose we need to recall and summarize the existence and uniqueness

of the conditional expectation inL1

cwk.E

s /.F/ [4,13,29] In particular, existenceresults for conditional expectation in Gelfand integration can be derived from themultivalued Dunford-Pettis representation theorem, see [4] A fairly general version

of conditional expectation for closed convex integrable random sets in the dual of

a separable Fréchet space is obtained by Valadier [29, Theorem 3] Here we needonly a special version of this result in the dual space Es

Theorem 5 Let  be a closed convex valued integrable random set in Es Let B

be a sub--algebra of F Then there exists a closed convex B-measurable mapping

On the Integration of Fuzzy Level Sets 13

Theorem5allows to obtain the weak compactness of the conditional expectation

of convex weakly compact valued integrably bounded mappings in E with strongseparable dual Indeed if F WD Eb is separable and if  is a convex weakly

compact valued measurable mapping in E with .!/  ˛.!/BE where ˛ 2 L1

R,

then applying Theorem4to Fgives †.!/ D EB.!/  E with †.!/ 

EB˛.!/BE  where BE  is the closed unit ball in E As S1

 is  L1

E; L1Ecompact, S1

Theorem 6 Given  2 L1

cwk.E

s /.F/ and a sub--algebra B of F, there

exists a unique (for equality a.s.) mapping† WD EB 2 L1

†.B/ denotes the set of

allL1EŒE.B/ selections of †) and satisfies

Now we need at first a conditional expectation version for Lemma1

Lemma 2 Let B be a sub--algebra of F and let X be a random fuzzy convex upper semicontinuous integrand X W   E

s ! Œ0; 1 with the following properties:

for all!2  and for all x 2 E.

LetEBX˛be the conditional expectation of the level setsL˛.X /WD X˛, ˛ 20; 1 Then we have

\

EBX˛k D EBX

˛:

On the Integration of Fuzzy Level Sets 15

Then the following hold

EBX˛.!/DT

k1EBX˛k.!/ for every ! 2  and every ˛ 20; 1, whenever

˛1< ˛2: : : < ˛k! ˛.

Proof Here we will use again the monotonicity of the conditional expectation and

the monotonicity of the level sets, namely for ˛  ˇ; Xˇ  X˛ and EBXˇ 

EBX˛ We have to check that

EBX˛.!/D\

k1

for every ! 2  whenever 0 < ˛1 < ˛2 < : : : < ˛k ! ˛ By the continuity

of the level sets (3) and the dominated convergence theorem for the conditionalexpectation, we have

for all x 2 E, so that the desired inclusion follows from the arguments developed

With the above considerations, we produce a general result on the conditionalexpectation of a convex weak-compact valued X˛ 2 L1

cwk.Es/.F/ depending onthe parameter ˛ 20; 1

Theorem 8 Let B be a sub--algebra of F, and let X W 0; 1  E

s be a convex weak-compact valued mapping with the following properties:

(1) jX.!; ˛/j  g 2 L1for all.!; ˛/2 0; 1,

(2) For each!2 , X.!; :/ is scalarly left continuous on 0; 1,

(3) For every fixed˛20; 1, X:.; ˛/ is scalarly F-measurable.

Then the convex weak-compact valued conditional expectation EBX˛ of the mappingX˛enjoys the properties

(a) For each!2 , ˛ 7! E BX

˛.!/ is scalarly left continuous on 0; 1, (b) For each˛20; 1, ! 7! E BX

˛.!/ is scalarly B-measurable on , (c) Assume further that˛7! X.!; ˛/ is decreasing, for every fixed !, i.e ˛ < ˇ 2

0; 1 implies X.!; ˇ/  X.!; ˛/, then 0 < ˛1< ˛2 < : : : < ˛k ! ˛ implies

(2) For each!2 , X.!; :/ is scalarly left continuous on 0; 1,

(3) For every fixed˛20; 1, X.:; ˛/ is scalarly F-measurable on .

16 C Castaing et al.

Then the convex weak-compact valued mappingEX˛ WD R

X.!; ˛/dP enjoys the properties

(a) ˛7! EX˛is scalarly left continuous on 0; 1,

(b) Assume further that˛ 7! X.!; ˛/ is decreasing, for every fixed !, then 0 <

˛1< ˛2< : : : < ˛k! ˛ implies EX˛DT

k1EX˛ k:

Corollary 2 Assume that E D Rd and that ;F; P / has no atoms Let X W

0; 1  Rdbe a compact valued mapping with the following properties: (1) jX.!; ˛/j  g 2 L1for all.!; ˛/2 0; 1,

(2) For each!2 , X.!; :/ is scalarly left continuous on 0; 1,

(3) For every fixed˛20; 1, X.:; ˛/ is scalarly F-measurable on .

Then the convex compact valued mapping EX˛ WD R

X.!; ˛/dP enjoys the properties

(a) ˛7! EX˛is scalarly left continuous on 0; 1,

(b) Assume further that˛ 7! X.!; ˛/ is decreasing, for every fixed !, then 0 <

representa-Lemma 3 Let.C˛/˛2Œ0;1be a family of convex weak-compact subsets inEwith the properties:

(1) C0WD E

s,

(2) Cˇ C˛;8˛ < ˇ 2 Œ0; 1,

(3) C˛ rBE for all˛20; 1,

(4) ˛7! ı.x; C˛/ is left continuous on 0; 1 for all x 2 E.

There is a unique fuzzy convex upper semicontinuous variable'W E

s ! Œ0; 1 such thatfx2 E

s W '.x/ ˛g D C˛, for every˛20; 1, where ' is given by

On the Integration of Fuzzy Level Sets 17

for all x 2 E Note that by (2) and (3), C˛ n/ is uniformly bounded and

decreasing Since the w-topology coincides with the metrizable topology m

on bounded subsets in the weakdual, by Theorem 5.4 in [5]

'.x/D supf˛ 2 Œ0; 1 W x2 C˛g  ˛0thus x2 Œ'  ˛0 To show the converse inclusion, let x2 Œ'  ˛0 Then

'.x/D supf˛ 2 Œ0; 1 W x2 C˛g  ˛0:

If '.x/ > ˛0, there exists ˛1 ˛0with x2 C˛1 But then we have C˛ 1  C˛0

by (2), thus x2 C˛0 Assume that '.x/D ˛0 Then there exists ˛k/ such that

x2 C˛k for each k and that ˛k" ˛0 But then

E.X / is the fuzzy expectation of X

Proof Apply Lemma3to the family C˛ DR

X˛dP /˛2Œ0;1with C0WDR

X0dP

D E

by taking account of Proposition3and Theorem4 

Now it is possible to provide the fuzzy conditional expectation of random fuzzy

convex upper semicontinuous integrand

Theorem 11 Let B be a sub--algebra of F and let X be a random fuzzy convex upper semicontinuous integrandX W   E

s ! Œ0; 1 with the following properties:

On the Integration of Fuzzy Level Sets 19

Proof Step 1. By Theorem6, recall that EBX˛2 L1

cwk.E

s /.B/ and by virtue ofTheorem7

EBX˛.!/D \

k1

EBX˛k.!/

for every ! 2  and every ˛ 20; 1, whenever ˛1< ˛2; : : : < ˛k! ˛

Step 2. Let ˛020; 1 and let ! 2  Let x2 EBX˛ 0.!/ Then

˛02 f˛ 2 Œ0; 1 W x2 EBX

˛.!/gwhich implies

'!.x/D supf˛ 2 Œ0; 1 W x2 EBX˛.!/g  ˛0thus x2 Œ'! ˛0 To show the converse inclusion, let x2 Œ'! ˛0 Then

'!.x/D supf˛ 2 Œ0; 1 W x2 EBX

˛.!/g  ˛0:

If '!.x/ > ˛0, there exists ˛1  ˛0with x 2 EBX

˛ 1.!/ But then we have

EBX˛ 1.!/  EBX˛0.!/ by the monotonicity of the conditional expectation,thus x 2 EBX

˛ 0.!/ Assume that '!.x/D ˛0 Then there exists ˛k/ suchthat x2 EBX

˛ k.!/ for each k and that ˛k" ˛0 But then

is a sub- -algebra ofF, EX is the expectation of X and E BX the conditional

expectation of X , then, for any B 2B, we haveRBEBXdP DR

BXdP , now wehave a similar equality if we deal with a random upper semicontinuous fuzzy convexintegrand X , fuzzy expectation QE.X / and fuzzy conditional expectation QE.XjB/.

Namely the following equality holds

20 C Castaing et al.

Our results can be applied to the convergence of convex weakly compact valuedlevel sets of a random upper semicontinuous integrand defined on a separablereflexive Banach space using the fuzzy expectation and the fuzzy conditionalexpectation

Next we will provide some SLLN results for fuzzy random variables in aseparable Banach space

Banach Space

Let c.E/ (resp k.E/) (resp cwk.E/) (resp ck.E/) denote the set of all nonempty

closed (resp compact) (resp convex weakly compact) (resp convex compact)subsets in E Here we focus on convergence in the Polish space ck.E/; dH/ where

dH is the Hausdorff distance on ck.E/ Let us recall and summarize some neededresults

Lemma 4 Let.Xn/ be a sequence in k.E/ If

Proof See e.g Arstein-Hansen [1], de Blasi and Tomassini [9], Hiai [12]

The following result is borrowed from Castaing and Raynaud de Fitte [6,Theorem 4.8]

Theorem 12 Let.Xn/ be a pairwise independent identically distributed sequence

of integrably bounded ck.E/-valued such that gWD supn2NjXnj  r is integrable, then

EŒXnD EŒX12 ck.E/; 8n 2 N

On the Integration of Fuzzy Level Sets 21

Remarks It is important to have the convexity and the norm compactness [2] ofEŒXn D EŒX1 because EŒX1 is the norm compactness limit in our SLLN It isalso worth to note that if Xn/ is a sequence of convex compact valued integrablybounded i.i.d random sets, the random variable g D supnjXnj is necessarilyconstant (with finite value) See [6] for details

Now we provide some applications to the SLLN for pairwise i.i.d compactvalued integrably bounded random sets

Theorem 13 Let.Xn/ be a pairwise independent identically distributed sequence

of integrably bounded k.E/-valued random sets in E such that gWD supn2NjXnj 

n

X

i D1

coXi; EŒcoX1/D 0 a.s

Invoking Lemma4yields

Theorem 14 AssumeED Rdand.;F; P / has no atom Let Xn/ be a pairwise independent identically distributed sequence of integrably bounded k.E/-valued random sets in E such that gWD supn2NjXnj  ˛ is integrable Then

22 C Castaing et al.

Proof Since coXn/ is pairwise independent identically distributed integrablybounded ck.E/-valued, Theorem12shows that

dH.1n

n

X

i D1

coXi; EŒcoX1/D 0 a.s

By invoking Lemma4it follows that

But E DRdand ;F; P / has no atom, thus

EŒXnD EŒcoXnD EŒcoX1D EŒX12 ck.Rd/;8n 2 N

Remark If E DRd, ;F; P / has no atom and X is compact valued integrably

bounded, then EŒX  is compact convex, this result is not valid in a separable Banachspace

Now is a version of SLLN in the primal space for fuzzy random variables

Theorem 15 Let.Xn/n2 Nbe a sequence of random fuzzy convex upper

semicon-tinuous variableXnW   E ! Œ0; 1 with the following properties:

(1) fx 2 E W Xn.!; x/ > 0g is compact, for each n 2 N and for each ! 2 , (2) g WD supnjL0C.Xn/j is integrable

Assume that

(3) XnD L˛.Xn//n2 Nis pairwise i.i.d, for each˛20; 1,

(4) Xn

˛ CD L˛C.Xn//n2 Nis pairwise i.i.d, for each˛2 Œ0; 1Œ.

Then we have, for every˛20; 1,

Assume further that the following condition is satisfied:

(5) Given " > 0, there exists a partition 0D ˛0 < ˛1 < : : : < ˛m D 1 of Œ0; 1 such that max1kmdH.EŒcoX1

˛ C k1

; EŒX1

˛ k/ < ".

Then we have

On the Integration of Fuzzy Level Sets 23

lim

n!1 sup

˛20;1

dH.1n

n

X

i D1

X˛i; EŒX˛1/D 0:

Proof (a) Since Xn/n2 N and X˛Cn /n2 N are independent identically distributed

compact valued random variables, we have, for every ˛ 20; 1, by Theorem13,

k/ < ":

Let ˛ 20; 1 Then there exist k (depending on ˛) such that ˛k1 < ˛  ˛k

We will use some elementary facts:

1n

n

X

i D1

X˛ik  1n

n

X

i D1

X˛i  1n

˛ EŒcoX1

˛ C k1:

Now we have the estimation

; EŒcoX1

˛ C k1

/C 2dH.EŒcoX1

˛ C k1

˛ C k1/C2 max

1kmdH.EŒcoX1

˛ C k1; EŒX˛1

k/WD I1C I2C I3:

24 C Castaing et al.From (8) it follows that

˛ C k1/D 0 a.s

for every k D 1; : : : ; m Hence for a.s ! 2 , we have

Finally we have

dH.1n

On the Integration of Fuzzy Level Sets 25Whence

lim

n!1 sup

˛20;1

dH.1n

n

X

i D1

X˛i.!/; EŒX˛1/D 0since " is arbitrary

Here is an important variant

Theorem 16 AssumeE D Rd and.;F; P / has no atom Let Xn/n2 N be a

sequence of random fuzzy convex upper semicontinuous variableXn W   E !

Œ0; 1 with the following properties:

(1) fx 2 E W Xn.!; x/ > 0g is compact, for each n 2 N and for each ! 2 , (2) g WD supnjL0C.Xn/j is integrable

n

X

i D1

X˛i; EŒX˛1/D 0 a.s.

Proof (a) Since X˛n/n2 N and X˛Cn /n2 N are pairwise independent, identically

distributed compact valued random variables, by Theorem14, we have

(b) Let " > 0 be given Using a technique similar to the one developed in Joo et

al [17, Theorem 3.1], we provide a partition 0 D ˛0< ˛1 < : : : < ˛m D 1 ofŒ0; 1 such that

max

1kmdH.EŒX1

˛ C k1; EŒX˛1k/ < ":

26 C Castaing et al.

Let ˛ 20; 1 Then there exist k (depending on ˛) such that ˛k1< ˛ ˛k Wewill use some elementary facts:

1n

n

X

i D1

X˛i  1n

:

EŒX˛1k EŒX1

˛ EŒX1

˛ C k1:

Now we have the estimation

˛ C k1

/C2dH.EŒX1

˛ C k1; EŒX˛1k/

We discuss in this section the concept of fuzzy martingale and integrand martingaleand provide some related convergence results Let Fn/n2N be an increasingsequence of sub  -algebras of F such that F is the -algebra generated by

[n2NFn Taking into account the results and notations developed in Sect.4, the

expected value (or expectation) of the fuzzy convex upper semicontinuous variable

Xncan be defined as a fuzzy variable QE.Xn/ such that the level set Œ QE.Xn/˛D Xn

for every ˛ 20; 1 and also the fuzzy conditional expectation of XnC1with respect

toFncan be defined as anFn˝ B.E/-measurable, upper semicontinuous fuzzy

convex integrand QE.XnC1jFn/ such that Œ QE.XnC1jFn/˛ D EFnXnC1

On the Integration of Fuzzy Level Sets 27

conditional expectation and by assuming that Xn takes its values in a subspace of

fuzzy sets u 2 FcL with the property that the function ˛ 7! L˛.u/ is Lipschitz

with respect to the Hausdorff distance dH.L˛.u/; Lˇ.u//  C j˛  ˇj, for every

˛; ˇ20; 1, where C is a positive constant

The above considerations lead to martingales depending on a parameter andintegrand martingales independently of the structure of fuzzy sets

Now we provide a result of existence of conditional expectation for normalintegrands on a separable Banach space E A mapping ‰ W   E ! R is a

F- normal integrand if it satisfies

(a) ‰.!; :/ is lower semicontinuous on E for all ! 2 ,

(b) ‰ isFNB.E/-measurable.

Let us recall a result of existence of conditional expectation for this class ofnormal integrands [3, Theorem 5.2] and [16]

Theorem 17 Let‰W   E ! R be a F-normal integrand satisfying

(i) There exista2 L1

E.;G ; P / and for all A 2 G Further, the integrand E G ‰ is unique

modulo the sets of the formN E, where N is a P -negligible set in G E G ‰ is the

conditional expectation of ‰ relative to G

Using the conditional expectation of normal integrands, we may define the notion

of lower semicontinuous integrand martingale as follows

Definition 2 Let Fn/n2Nbe an increasing sequence of sub  -algebras ofF such

thatF is the -algebra generated by [n2NFnand ‰nW   E ! RC.n2 N/ be a

for all A 2Fn, for all u 2 L1E.;Fn; P / and for all n 2N:

Now we provide an epiconvergence result for integrand martingales

28 C Castaing et al.

Theorem 18 Let.Fn/n2Nbe an increasing sequence of sub -algebras of F such that F is the -algebra generated by [n2NFn,‰ W   E ! RC a F-normal integrand such that ‰.:; u.:// is integrable for all u 2 L1

E.;F; P / Let E Fn‰.n 2 N/ be the conditional expectation of ‰ relative to Fn whose existence is given by Theorem 17 Then, for each u 2 L1

E.;F; P / the following variational inequality holds:

Let u 2 L1E.;F; P / Let p 2 N Since E is separable, applying the measurable

selection theorem [8, Theorem III-22], it is not difficult to provide anF-measurable

mapping vk;p;uW  ! E such that

0 ‰.!; vk;p;u.!//C kjju.!/  v k;p;u.!/jjE ‰k.!; u.!//C 1

pfor all ! 2 , so that ! 7! ‰.!; vk;p;u !// and ! 7! kjju.!/  v k;p;u.!/jjE areintegrable, and so vk;p;u2 L1

E.;F; P / By our assumption, ! 7! ‰.!; v k;p;u.!//

is integrable, too Applying Lévy’s theorem yields a negligible set Nk;p;usuch that,for all ! … Nk;p;u,

On the Integration of Fuzzy Level Sets 29

Set NuWD [k2N;p2NNk;p;u Then Nuis negligible Taking the supremum on k 2N

in the extreme terms yields

Theorem 19 LetXnW 0; 1 ! RCwith the properties

(a) Xn.:; ˛/ is Fn-measurable for alln2 N, for all ˛ 2 Œ0; 1,

(b) jXn.!; ˛/ Xn.!; ˇ/j  C.!/j˛  ˇj, for all n 2 N, for all ! 2 , for all

˛; ˇ2 Œ0; 1, where C is a positive integrable function,

(c) 0  Xn.!; ˛/  .!/, for all n 2 N, for all ! 2 , where  is a positive integrable function,

(d) For each˛2 Œ0; 1, X˛/D Xn.:; ˛// is a martingale.

Then there exists anL1

2C C-boundedRC-valued C -Lipschitz integrand X1W  Œ0; 1! R satisfying

Œ0; 1, so that there is a negligible set N such that

jY˛

1.!/ Yˇ

1.!/j  C.!/j˛  ˇjj 8! 2  n N

30 C Castaing et al.for all ˛; ˇ 2 Q Let us set for all !; ˛/ 2   Q, Z1.!; ˛/ D Y˛

L1

2C C-boundedRC-valued C -Lipschitz integrand, because

0 X1.!; r/ C.!/jr ˛jCZ1.!; ˛/ 2C.!/CZ1.!; ˛/ 2C.!/C.!/for all !; r 2   Œ0; 1 Now we prove that X1satisfies the required convergence

By the above construction and hypothesis the sequence

jX˛

n  Xˇ

nj  C.!/j˛  ˇjand

jX˛

1 Xˇ

1j  C.!/j˛  ˇjfor all ˛; ˇ 2 Œ0; 1 By the Lipschitz property of X1and Jensen’s inequality, wehave the estimation

jEFnX1˛  EFnX1ˇj  EFnjX˛

1 Xˇ

1j  EFnCj˛  ˇjfor all ˛; ˇ 2 Œ0; 1 Consequently the mapping ˛ 7! Xn˛ EFnX˛

1is Lipschitz onŒ0; 1 So we conclude that

On the Integration of Fuzzy Level Sets 31Finally we get

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Adv Math Econ 19, 33–55 (2015)

A Theory for Estimating Consumer’s Preference from Demand

Yuhki Hosoya

Abstract This study shows that if the estimate error of a demand function

satisfying the weak axiom of revealed preference is sufficiently small with respect tolocal C1topology, then the estimate error of the corresponding preference relation(which is possibly nontransitive, but uniquely determined from demand function,and transitive under the strong axiom) is also sufficiently small Furthermore, weshow a similar relation for the estimate error of the inverse demand function withrespect to the local uniform topology These results hold when the consumptionspace is the positive orthant, but are not valid in the nonnegative orthant

Keywords Demand function • Inverse demand function • Integrability theory •

Closed convergence topology • Uniform convergence topology • C1convergencetopology

Article type: Research Article

34 Y Hosoya

In economics, estimating a preference relation from observed data is very importantfor conducting welfare analysis Both revealed preference theory and integrabilitytheory approach this problem, and share the same key idea Because purchasebehaviors are observed, the demand function is much easier to estimate than thepreference relation Hence, if there exists a method to calculate the preferencerelation from the demand function, then the difficulty of estimating the preferencerelation decreases

In the nineteenth century, Antonelli [2] presented a sufficient condition for thelocal existence of a utility function Pareto [10] also considered this problem,before Samuelson [14] connected their classical conditions to the symmetry of theSlutsky matrix of the demand function The general existence of the correspondingpreference of the demand function obeying the strong axiom of revealed preferencewas proved by Richter [13] and Afriat [1] For computations, Hurwicz and Uzawa[7] presented a method of constructing a utility function from a demand function.Kim and Richter [9] and Quah [12] extended the result of Richter [13] for demandfunctions obeying the weak axiom of revealed preference

This study is based on the work of Hosoya [6], who presented a method ofconstructing a preference relation for a smooth demand function with the weakaxiom, whose demand function is exactly the same as the original demand function.This preference relation is complete, but may be nontransitive.1 However, thetransitivity is revived when this demand function obeys the strong axiom Moreover,this study shows that such a preference is “unique”, i.e., there is no other usual(that is, complete, continuous, and “p-transitive”, which is defined later) preferencerelation corresponding to the same demand function Let P W f 7!%f denotethis mapping Thus, if we obtain an estimate f of the demand function, then wesimultaneously obtain an estimate %fD P.f / of the corresponding preferencerelation

However, the actual estimation process is always associated with the estimate error If the estimator uses a sophisticated econometric method, then the estimate

error of the demand function may become small However, it remains unknownwhether the estimate error of the corresponding preference relation is also reduced

To deal with this problem, we should first clarify the meaning of the term “small”

In general, economists use the closed convergence topology for the space of the

preference relations.2 Hence, the sentence “the estimate error of the preferencerelation is small” means that the estimated preference is within a smallneighborhood

1 Therefore, in this paper the notion of a “preference relation” does not include the transitivity requirement.

2 This topology was induced by Kannai [ 8 ] For a detailed treatment of this topology, see Hildenbrand [ 5 ].